WKB Method for Swaption Smile

Transcription

1 WKB Method for Swaption Smile Andrew Lesniewski BNP Paribas New York February Abstract We study a three-parameter stochastic volatility model originally proposed by P. Hagan for the forward swap rate. The model is essentially a stochasticied version of the CEV model where the volatility parameter is itself a stochastic process. We construct a computationally efficient asymptotic solution to this model. This solution allows one to fit a variety of shapes of the volatility smiles in the swaption markets. The technique used to obtain this solution is a WKB expansion around geodesic motion on a suitable hyperbolic manifold. Contents Introduction 2 2 Stochastic CEV model 4 3 Exactly solvable case 6 4 Geometry of the full model 7 5 WKB method 9 Transparencies for the talk at the Mathematical Finance Seminar at the Courant Institute of Mathematical Sciences New York

2 6 Probability distribution 7 Further developments 3 References [] Hagan P.: Stochastic β models Paribas 998 [2] Lesniewski A.: Stochastic CEV models and the WKB expansion BNP Paribas 200 [3] Molchanov S. A.: Diffusion processes and Riemannian geometry Russian Math. Surveys [4] Freidlin M.: Markov Processes and Differential Equations: Asymptotic Problems Birkhauser 996 Introduction A swaption is an option to enter into a swap. A receiver payer swaption gives the owner the right to receive pay fixed rate on the swap. Market lingo: a T into M swaption is a swaption expiring T years from now on an M year swap. Thus a 7% 5 into 0 receiver is an option to receive 7% on a 0 year swap starting 5 years from now. The market practice is to quote prices and calculate risk parameters of European swaptions in terms of Black s model: df t = σf t dw t where F t is the forward swap rate and σ is the implied volatility or Black s volatility. For simplicity rather than pricing calls and puts we consider the Arrow-Debreu security whose payoff at time T is given by δ F T F where δ denotes Dirac s delta function. The time t price G t f; T F of this security or the transition probability is the solution to the following parabolic problem: G t + 2 σ2 f 2 2 G f 2 = 0 G t f; T F = δ f F at t = T. 2

3 Thus the price V of a payer struck at K is V = N G t f; T F max F K 0 df where N is depends on the notional principal of the transaction and today s term structure of rates but not σ. The solution is G t f; T F = f 2πτF 3 σ 2 exp log f/f 2 2τσ 2 τσ2 8 where τ = T t. This leads to the well known Black s formula for pricing swaptions. The reality of the market is that implied volatility is a function σ = σ K T f of strike K time to expiry T today s value of the forward f. The dependence of σ on K reflects economic realities and is referred to as the option smile. Why is modeling smile important? Transaction pricing; Portfolio mark to market; Portfolio risk: smile adjusted delta is = V f + V σ σ f = 0 + Λ σ f where 0 is the B-S delta and Λ is the B-S vega. Similarly for the other greeks; 3

4 Calibrating term structure models for pricing exotic structures transactions with embedded options etc. How to model smile? Interpolate brokers quotes; Shifted lognormal model: df t = σ F t + σ 0 dw t ; CEV model df t = σf β t dw t 0 β ; Stochastic volatility models;... Implementation issues: Exact solutions; Numerical implementations tree MC PDE; Approximate analytic solutions. 2 Stochastic CEV model Replace Black s model by the system df t = σ t b F t dw t dσ t = vσ t dz t where the two Brownian motions are correlated: E [dw t dz t ] = ρdt. 4

5 For the CEV model b F = F β. Hagan s formula This stochastic volatility model was originally studied by P. Hagan who derived a very useful approximate expression for the implied volatility. We state this result in terms of the implied normal volatility from which Black s volatility can be easily calculated. Implied normal volatility σ n is has the following asymptotic expansion: σ n = v f K δ 0 f K where + 24 [ 2γ2 γ 2 σ 2 b f 2 + 8ρvσγ b f + 2 3ρ 2 v 2 τ ] +... ζ2 2ρζ + + ζ ρ δ 0 ζ = log ρ ζ = v f du σ b u K and τ = T t γ = b f γ 2 = b f. Our goal is to explain the origin of this formula in a general setting which is applicable to a large class of models. We consider the Arrow-Debreu security whose payoff at time T is given by δ F T F σ T Σ where δ denotes Dirac s delta function. The time t price G t f σ; T F Σ of this security is the solution to the following parabolic problem: G t + 2 σ2 b f 2 2 G f + 2vρb f 2 G 2 f σ + 2 G v2 σ 2 = 0 G t f σ; T F Σ = δ f F σ Σ at t = T. The price of a payer struck at K is now V = N G t f σ; T F Σ max F K 0 df dσ = N G σ t f; T F max F K 0 df 5

6 where G σ t f; T F = G t f σ; T F Σ dσ is the marginal transition probability. The coefficients in this problem are time independent and so G is a function of τ = T t. Denote this function by G f σ F Σ; τ and introduce the following notation: s = τ/t x = f X = F y = σ/v Y = Σ/v K x y X Y ; s = vt G x vy X vy ; st. In terms of these variables the initial value problem can be recast as: K = s 2 εy2 b x 2 2 K x + 2ρb x 2 K 2 x y + 2 K y 2 K x y X Y ; 0 = δ x X y Y at s = 0 where ε = v 2 T. It will be assumed that ε is small and it will serve as the parameter of our expansion. The heuristic picture behind this idea is that the volatility varies slower than the forward and the rates of variability of f and σ/v are similar. The time T defines the time scale of the problem and thus s is a natural dimensionless time variable. Expressed in terms of the new variables our problem has a natural differential geometric content which is key to its solution. 3 Exactly solvable case Let b F = and ρ = 0: df t = σ t dw t dσ t = vσ t dz t E [dw t dz t ] = 0. 6

7 Also define x = f y = σ/v. Then the problem becomes: K = 2 K τ 2 y2 x + 2 K 2 y 2 K x y X Y ; τ = δ x X y Y at τ = 0. This is the heat equation on the Poincare plane! Recall that the Poincare plane is the upper half plane H 2 = {x y : y > 0} with the line element ds 2 = dx2 + dy 2 y 2. This comes from the metric tensor h = y The geodesic distance d z Z between two points z Z H 2 z = x + iy Z = X + iy is z Z 2 cosh d z Z = + 2yY where z Z is the Euclidean distance between z and Z. The heat equation on the Poincare plane can be solved in closed form: K z Z; τ = e d2 /2τ 2 4πτ 3/2 Asymptotic expansion as τ 0: K z Z; τ = /2τ 2πτ e d2 d sinh d dzz. ue u2 /4τ cosh u cosh d z Z du d coth d + 8 d 2 4 Geometry of the full model τ + O τ 2. Let M 2 denote the first quadrant {x y : x > 0 y > 0} and let g denote the metric: g = ϱ 2 ρ y 2 bx 2 y 2 ρ bx y 2 bx y 2. 7

8 The case of ρ = 0 and b x = reduces to the Poincare metric. The metric g is the pullback of the Poincare metric under a diffeomorphism: choose p > 0 and define φ p : M 2 H 2 by φ p z = ϱ 2 x where z = x y. The Jacobian φ p of φ p is φ p z = ϱ 2 bx 0 p du b u ρy y and so φ ph = g where φ p denotes the pullback of φ p. The manifold M 2 is thus isometrically diffeomorphic with a submanifold of the Poincare plane. Consequently we have an explicit formula for the geodesic distance δ z Z on M 2 : ρ ϱ 2 cosh δ z Z = cosh d φ p z φ p Z x du X bu = + 2 2ρ y Y x X 2 ρ 2 yy du bu + y Y 2 where z = x y and Z = X Y are points on M 2. The volume element on M 2 is given by dxdy ϱ 2 b x y. 2 Let z = x z 2 = y and let µ = / z µ µ = 2 denote the corresponding partial derivatives. We denote the components of g by g µν and use g and g to raise and lower the indices: z µ = g µν z ν µ = g µν ν = / z µ where we sum over the repeated indices. Explicitly = y 2 b x 2 + ρb x 2 2 = y ρb x. Consequently the initial value problem can be written as: s K z Z; s = 2 ε µ µ K z Z; s K z Z; 0 = δ z Z. Except for b x = µ µ is not the Laplace-Beltrami operator on M 2. 8

9 5 WKB method We seek K z Z; s in the form K z Z; s = R z Z; s exp 2πε ε S z Z; s S z Z; s is assumed independent of ε. R z Z; s depends on ε and is assumed smooth at ε = 0. Substituting we obtain the following two PDEs: S s + 2 µ S µ S = 0 where the subscript s denotes the derivative with respect to s and R 2 s + µ R 2 µ S = εr µ µ R. Equation is the Hamilton-Jacobi equation for a free motion of a particle on M 2 S z Z; s is the action function. Now factor out the ε-independent part of R z Z; s: R z Z; s = q z ZQ z Z; s and make the following asymptotic expansion in ε: Q z Z; s = k 0 ε k Q k z Z; s with Q 0 z Z; s =. Then the function q z Z; s satisfies the transport equation q s + µ q µ S = 0 and for Q z Z; s we find the equation: Q s + µ Q µ S = ε 2 q µ µ qq. This last equation is equivalent to an infinite system: Q k+ s + µ Q k+ µ S = 2 q µ µ qq k which we call the WKB hierarchy. The three equations: 9

10 the Hamilton-Jacobi equation the transport equation the WKB hierarchy form a basis for constructing an asymptotic solution to our problem. Solve the Hamilton-Jacobi equation We set S z Z; s = S 0 φ z φ Z ; s where φ = φ X. Then S 0 satisfies the Hamilton-Jacobi equation on the Poincare plane: S 0 s + 2 y2 S 0 x 2 + S 0 y 2 = 0. Seek a spherically symmetric solution S 0 w W ; s = f r; s where r = d w W. Then f satisfies f s + 2 f 2 r = 0 and so f r; s = 2s r2 + const. We will choose const = 0. Consequently S z Z; s = d φ z φ Z2 2s = δ z Z2 2s It is after reinstating the constant const the complete integral of the Hamilton- Jacobi equation. Solve the transport equation We set q z Z; s = p φ z φ Z ; s det φ z = p φ z φ Z ; s ϱ2 b x to find that p s + µ p µ S 0 = y 2 log B S 0 p 0

11 where B φ z = b x. Substituting p = q 0 B we find q qs 0 + y 2 0 Sx 0 + q 0 S 0 x y = 0. This is the transport equation on the Poincare plane. solution q 0 = f r s. Then y We seek a radial s fr sinh r s + r fr sinh r r = 0. This means that f r s r sinh r is a function of r/s: and so q z Z; s = f r s = χ r/s r sinh r ϱ 2 B x X b x χ δ z Z /s δ z Z sinh δ z Z. Solve the WKB hierarchy Well ok let s stick with Q 0 z Z; s =. 6 Probability distribution We put everything together and verify that in order for the initial condition to be satisfied we need to choose χ u = Cu 2. Hence finally we obtain the asymptotic formula B x X K z Z; s = 2πεs ρ 2 b x { δ z Z sinh δ z Z exp } δ z Z2 + O sε. 2εs First we calculate the approximate marginal probability distribution from the WKB Green s function: K y x X; s = = K z Z; s dy 0 2πεs B x X ρ 2 b x 0 δ z Z sinh δ z Z exp δ z Z2 dy 2εs Y 2

12 To evaluate this integral we use the Laplace method: as ε 0 f u e φu/ε 2πε du = φ u 0 e φu 0/ε f u 0 + O ε 0 if u 0 is the unique minimum of φ with φ u 0 > 0. Let us introduce the notation: ζ = y x X du b u. Given x X and y let Y 0 be the value of Y which minimizes the distance δ z Z and let δ 0 ζ be the corresponding value of δ z Z. Explicitly Introduce the notation Y 0 ζ y = y ζ 2 2ρζ + ζ2 2ρζ + + ζ ρ δ 0 ζ = log. ρ I ζ = ζ 2 2ρζ +. This yields K y x X; s = { } B x X exp δ2 0 + O sε 2πεsy 2 I ζ 3 b x 2εs This is the desired asymptotic form of the marginal probability distribution. Let us now compare this result with the normal distribution function: x X2 P x X; s = exp 2πsεy 2 n 2sεyn 2 where y n is related to the normal volatility σ n by y n = σ n /ε /2. We shall relate the cumulative distribution function } P x X; s dx = exp { h2 dh K 2πεs K 2εs = K x 2 erfc 2sεyn 2

13 to the cumulative distribution function of K y x X; s. Neglecting the terms of order O ε 2 we have: { } b B x X K y x X; s dx = 2πεsy 2 I ζ 3 b x exp δ2 0 dx. 2εs K We substitute a new variable in the integral above x du h = h X = δ 0 y b u which yields K K y x X; s dx = K 2πεs hk X } B x X h exp { h2 dh I ζ b x 2εs Expanding as ε 0 and comparing the terms we obtain Hagan s formula. 7 Further developments Mean reversion in the volatility dynamics df t = σ t b F t dw t dσ t = λσ t log σ σ t dt + vσ t dz t E [dw t dz t ] = ρdt. Heston s model of stochastic volatility Time dependent parameters df t = a t b F t σ t dw t dσ t = v t σ t dz t E [dw t dz t ] = ρ t dt. To analyze this model one will also need other asymptotic techniques like homogenization. Term structure model with stochastic volatility consistent with the vanilla model. 3